If \( g(x) \) is a polynomial with real coefficients and zeros of 2 (multiplicity 2 ), -5 (multiplicity 2 ), \( 7 i \), and \( 3+6 i \), what is the minimum degree of \( g(x) \) ? The minimum degree of \( g(x) \) is \( \square \),

To determine the minimum degree of \( g(x) \), we need to account for all the zeros and their multiplicities. The polynomial has the following zeros: - 2 with multiplicity 2 contributes 2 to the degree. - -5 with multiplicity 2 contributes 2 to the degree. - \( 7i \), a complex zero, requires its conjugate \( -7i \) to also be a zero, contributing 2 to the degree. - \( 3 + 6i \), another complex zero, requires its conjugate \( 3 - 6i \) to also be a zero, contributing 2 to the degree. Adding these contributions together, we have: \[ 2 (for\: 2) + 2 (for\: -5) + 2 (for\: 7i \: and \: -7i) + 2 (for\: 3 + 6i \: and \: 3 - 6i) = 8 \] Thus, the minimum degree of \( g(x) \) is \( \boxed{8} \).